Applied Statistics and Probability for Engineers

8-6 A PREDICTION INTERVAL FOR A FUTURE OBSERVATION 269

has a standard normal distribution. Replacing with Sresults in

which has a tdistribution with n1 degrees of freedom. Manipulating Tas we have done previ-

ously in the development of a CI leads to a prediction interval on the future observation Xn 1.

T

Xn 1 X

S

B

1 

1

n

The prediction interval for Xn 1 will always be longer than the confidence interval for 

because there is more variability associated with the prediction error than with the error of es-

timation. This is easy to see because the prediction error is the difference between two random

variables (Xn 1  ), and the estimation error in the CI is the difference between one random

variable and a constant ( ). As ngets larger ( ),the length of the CI decreases to

zero, essentially becoming the single value , but the length of the prediction interval

approaches 2z 2 . So as nincreases, the uncertainty in estimating goes to zero, although

there will always be uncertainty about the future value Xn 1 even when there is no need to

estimate any of the distribution parameters.

EXAMPLE 8-8 Reconsider the tensile adhesion tests on specimens of U-700 alloy described in Example 8-4.

The load at failure for n22 specimens was observed, and we found that 13.71 and

s3.55. The 95% confidence interval on was 12.14 15.28. We plan to test

a twenty-third specimen. A 95% prediction interval on the load at failure for this specimen is

Notice that the prediction interval is considerably longer than the CI.

EXERCISES FOR SECTION 8-6

6.16X 23 21.26

13.71 1 2.080 2 3.55

B

1 

1

22

X 23 13.71 1 2.080 2 3.55

B

1 

1

22

xt 2, n 1 s

B

1 

1

nXn^1 xt^ 2,n^1 s^ B^1 

1

n

x

X nS

X

A 100(1 )% prediction interval on a single future observation from a normal

distribution is given by

xt 2,n 1 s (8-29)

B

1 

1

nXn^1 ^ xt^ 2,n^1 s^ B^1 

1

n

Definition

8-49. Consider the tire-testing data described in Exercise 8-22.

Compute a 95% prediction interval on the life of the next tire of

this type tested under conditions that are similar to those em-

ployed in the original test. Compare the length of the prediction

interval with the length of the 95% CI on the population mean.

8-50. Consider the Izod impact test described in Exercise 8-23.

Compute a 99% prediction interval on the impact strength of

the next specimen of PVC pipe tested. Compare the length of

the prediction interval with the length of the 99% CI on the

population mean.

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